Industry literature related to drilling vibration modeling includes teachings directed to forced vibration (induced excitation) frequency-domain computational models with excitation at two or more frequencies. Drilling performance metrics determined by these models include vibration index values that relate the system response to the system excitation. Vibration indices and related discussions are discussed in part, for example, in U.S. Pat. No. 9,483,586 B2, U.S. Pat. No. 8,589,136 B2, and U.S. Pat. No. 8,977,523 B2. These models and disclosures do not inherently provide means to combine different excitation frequencies to represent a drilling operation.
Frequency-domain vibration models are computationally efficient and can be used to great benefit in drilling applications. For example there are many frequency-domain models in the literature for each of axial, lateral, and torsional vibrations. Different models have different boundary conditions, coupling, modes of vibration, element types, and so forth, but one common characteristic of linear forced vibration models is that the system excitation and response output occurs at the same frequency, and the output is linearly proportional to the input. In most cases, the amplitude of the input excitation is arbitrarily selected to be a reference value and may be constant for each excitation frequency.
Fourier analysis can be used to determine the frequency, or “spectral”, content of a time series of data, and a complete description includes both real and imaginary parts, or equivalently magnitude and phase. Those skilled in the art appreciate how various windowing processes and averaging of spectral calculations applying Fourier analysis can be used to estimate a spectrum of a time series that may be longer than the duration of a single Fourier calculation. The “periodogram” function available in the MATLAB program from The MathWorks provides such functionality. This function calculates the amount of signal energy present at each frequency for an extended time series of data. These amplitude (or magnitude) factors provide quantitative information on the contribution of each frequency to the resulting signal.
In any design process, a greater number of different criteria increases the complexity of the selection process as it becomes increasingly difficult to meet all criteria as the number to be satisfied increases. It is simplest to have as few design criteria as possible that meet the design objectives. The cited frequency domain models do not inherently provide weighting of results according to magnitude, indeed the calculations typically assume a reference input value and provide system response for this input but do not specify the relative contributions of each frequency. Time domain modeling, though computationally intensive, does provide the combination of these components in the model output.